What is U6 unemployment rate ?

The U6 unemployment rate counts not only people without work seeking full-time employment (the more familiar U-3 rate), but also counts "marginally attached workers and those working part-time for economic reasons."
Note that some of these part-time workers counted as employed by U-3 could be working as little as an hour a week.
And the "marginally attached workers" include those who have gotten discouraged and stopped looking, but still want to work.
The age considered for this calculation is 16 years and over

About variuos measures of unemployment

The Bureau of Labor Statistics measures employment and unemployment (of those over 16 years of age) using two different labor force surveys conducted by the United States Census Bureau (within the United States Department of Commerce) and/or the Bureau of Labor Statistics (within the United States Department of Labor) that gather employment statistics monthly. The Current Population Survey (CPS), or "Household Survey", conducts a survey based on a sample of 60,000 households. This Survey measures the unemployment rate based on the ILO definition.
The data are also used to calculate 5 alternate measures of unemployment as a percentage of the labor force based on different definitions noted as U1 through U6:

U1 : Percentage of labor force unemployed 15 weeks or longer.

U2 : Percentage of labor force who lost jobs or completed temporary work.

U3 : Official unemployment rate per ILO definition.

U4 : U3 + "discouraged workers", or those who have stopped looking for work because current economic conditions make them believe that no work is available for them.

U5 : U4 + other "marginally attached workers", or "loosely attached workers", or those who "would like" and are able to work, but have not looked for work recently.

U6 : U5 + Part time workers who want to work full time, but cannot due to economic reasons.

Below is the overview of these six measures.

U1:

This is the proportion of the civilian labor force that has been unemployed for 15 weeks or longer. This unemployment rate measures workers who are chronically unemployed. During business-cycle expansions, this rate captures structural unemployment. However, during lengthy business-cycle contractions, this rate is also likely to include a significant amount of cyclical unemployment. U1 tends to be relatively small, in the range of 1-2 percent.

U2:

This is the proportion of the civilian labor force that is classified as job losers (workers who have been involuntarily fired or laid off from their jobs) and people who have completed temporary jobs. During business-cycle expansions, this rate is likely to capture some degree of frictional unemployment. However, during business-cycle contractions, this rate is most likely to consist of cyclical unemployment. U2 is larger than U1, but still remains substantially less than the official unemployment rate (U3).

U3:

This is the official unemployment rate, which is the proportion of the civilian labor force that is unemployed but actively seeking employment.

U4:

This is the official unemployment rate that is adjusted for discouraged workers. In other words, discouraged workers are treated just like other workers who are officially classified as unemployed, being included in both the ranks of the unemployed and the labor force. It is technically specified as the proportion of the civilian labor force (plus discouraged workers) that is either unemployed but actively seeking employment or discouraged workers. The addition of discouraged workers generally adds a few tenths of a percentage point to the official unemployment rate.

U5:

This augments U4 by including marginally-attached workers to the unemployment rate calculation. Marginally attached workers are potential workers who have given up seeking employment for various reasons. One of these reasons is that the workers believe such effort would be futile, which places them in the discouraged worker category. Those who have other reasons for not seeking employment are placed in the broader marginally-attached workers category. The addition of marginally-attached workers adds a few more tenths of a percentage point to the official unemployment rate.

U6:

This augments U5 by including part-time workers to the unemployment rate calculation. The addition of part-time workers adds a full 2-3 percentage points to the official unemployment rate. This measure of unemployment is perhaps the most comprehensive measure of labor resource unemployment available.

Who is counted as unemployed?
Persons are classified as unemployed if they do not have a job, have actively looked for work in the prior 4 weeks, and are currently available for work. Actively looking for work may consist of any of the following activities:

Contacting:

An employer directly or having a job interview

A public or private employment agency

Friends or relatives

A school or university employment center

Sending out resumes or filling out applications

Placing or answering advertisements

Checking union or professional registers

Some other means of active job search

Who is not in the labor force?
Labor force measures are based on the civilian noninstitutional population 16 years old and over. Excluded are persons under 16 years of age, all persons confined to institutions such as nursing homes and prisons, and persons on active duty in the Armed Forces. The labor force is made up of the employed and the unemployed. The remainder—those who have no job and are not looking for one—are counted as "not in the labor force." Many who are not in the labor force are going to school or are retired. Family responsibilities keep others out of the labor force.